Unsymmetric Serendipity finite elements: Performance analysis

This paper presents a comparative analysis of the conventional finite element method (FEM) and the unsymmetric finite element method (UFEM) for Serendipity elements ( p ≤ 3 ), focusing on two factors: (i) achievable accuracy and (ii) computational costs. The UFEM, based on a Petrov–Galerkin formulation, uses metric shape functions as trial functions and parametric shape functions as test functions. This unique approach enhances the resistance against mesh distortion, as it ensures polynomial completeness of the Ansatz space of unsymmetric finite elements. Hence, higher accuracy can be achieved in complex geometries. However, the unsymmetric nature of UFEM leads to increased computational costs as a result of the added complexity of solving the resulting system of equations. This study provides a quantitative evaluation of the computational burden associated with achieving specific error thresholds for both methods. By analyzing a range of benchmark problems, we identify scenarios in which each method performs optimally, offering practical insights for selecting the appropriate approach based on accuracy demands and computational constraints. Our findings suggest that, while UFEM can produce superior accuracy, its computational efficiency depends on application-specific requirements and available resources.